Bayesian Surrogate Optimization Outperforms Deep Learning in Turbomachinery Design Under Severe Simulation Limits

Aerodynamic shape optimization for advanced aeronautic engine components achieves superior computational efficiency when pairing linear autoregressive Gaussian process surrogates with dimensionality reduction rather than deep learning frameworks under severe high-fidelity evaluation constraints, according to an engineering study published in Advanced Modeling and Simulation in Engineering Sciences. Researchers Marc Schouler, Anca Belme, and Paola Cinnella demonstrated that when computational budgets permit only ten high-fidelity simulations during aerodynamic design exploration, multi-objective Bayesian optimization assisted by an AR1 co-kriging surrogate and Proper Orthogonal Decomposition consistently surpassed an active-learning genetic algorithm driven by a multi-fidelity deep neural network. The investigation addressed the challenge of optimizing turbomachinery profiles when high-fidelity numerical solvers are thousands of times more expensive than low-fidelity evaluations. Across targeted aerodynamic metrics, the reduced Bayesian strategy delivered superior convergence and convergence speed, identifying blade geometries that reduced design-point pressure losses by up to seventeen percent while decreasing off-design losses by thirty percent.

Aerodynamic shape optimization plays a pivotal role in the design of next-generation aircraft and turbomachinery systems, where tightening environmental constraints demand increasingly aggressive component efficiency. Capturing critical flow physics, such as laminar-to-turbulent transition and separation bubbles, necessitates moving toward high-fidelity scale-resolving or transition-sensitive flow models. However, the extreme computational expenditure of these sophisticated numerical solvers severely restricts their deployment within automated design loops. When evaluating complex geometries, gradient-based optimization algorithms that depend on adjoint sensitivity formulations encounter severe practical obstacles, including the differentiation of legacy computational fluid dynamics software, excessive memory overhead, and irregular or chaotic gradient behavior in turbulence-resolving regimes. Consequently, engineers frequently rely on gradient-free approaches such as evolutionary algorithms and particle swarm optimization. While robust against numerical noise and multimodal response surfaces, gradient-free optimizers typically require thousands of evaluations, rendering direct coupling with high-fidelity solvers computationally prohibitive under strict industrial budget constraints.

To reconcile the high evaluation demands of gradient-free algorithms with limited computational resources, multi-fidelity surrogate modeling strategies construct lightweight statistical approximations of objective functions by fusing abundant, low-cost simulations with scarce, highly accurate evaluations. In typical aerodynamic applications, the cost ratio between low-fidelity and high-fidelity solvers ranges between ten and one hundred, allowing optimization frameworks to afford dozens or hundreds of high-fidelity queries. In contrast, this study evaluated an extreme cost imbalance regime where high-fidelity evaluations are up to ten thousand times more expensive than low-fidelity models, restricting the usable high-fidelity budget to roughly ten to twenty total evaluations. Under such severe constraints, standard surrogate methods face the curse of dimensionality, as high-dimensional geometric representations require substantial training samples to maintain predictive accuracy. The research team consequently formulated and compared two distinct surrogate-assisted architectures designed to extract maximal information from very few high-fidelity evaluations through structured infill sampling and latent-space dimension reduction.

The first optimization strategy implemented a multi-objective Bayesian optimization architecture based on Kennedy and O’Hagan’s linear autoregressive co-kriging framework, designated as the AR1 model, implemented through the open-source smt library. This formulation models the high-fidelity objective function as a scaled low-fidelity Gaussian process supplemented by an additive Gaussian process bias term that captures residual discrepancies. Because the nested design of experiments structure allows closed-form analytical representations of predictive variance, the AR1 model provides explicit statistical uncertainty estimates across the design space. To guide sequential sampling, the authors employed the minimum probability of improvement criterion within a regularized acquisition function, balancing local exploitation against global exploration. Each adaptive infill iteration selected one high-fidelity sample alongside two low-fidelity evaluations targeted at minimizing the lower confidence bound of each objective, supplemented by distance-maximization criteria solved via particle swarm optimization to maintain spatial diversity across candidate evaluations.

The competing strategy combined a multi-objective non-dominated sorting genetic algorithm, NSGA-II, with a multi-fidelity deep neural network implemented in PyTorch. The neural architecture approximated the target high-fidelity response by combining a linear sub-network and a non-linear sub-network scaled by a correlation coefficient, processing input coordinates alongside low-fidelity neural network predictions. Unlike the Gaussian process formulation, the neural network surrogate functions deterministically and scales readily to arbitrary objective dimensions without requiring independent surrogate models for each aerodynamic criterion. However, the deterministic nature of neural networks precludes Bayesian uncertainty-driven acquisition functions. Instead, the framework executed intermediate genetic algorithm optimizations across the surrogate surface at each iteration, selecting optimal trade-off candidates from the predicted Pareto set to compute with the high-fidelity solver while allocating additional low-fidelity queries to individual objective optima and distant unexplored coordinates.

Geometric parametrization of the aerodynamic profiles utilized Free-Form Deformation, embedding the two-dimensional airfoil within a controllable bounding lattice governed by eight movable control points. To mitigate the curse of dimensionality and maximize surrogate training efficiency from minimal data, the authors applied Proper Orthogonal Decomposition to the geometric coordinates generated through Latin hypercube sampling. By constructing a correlation matrix from geometric fluctuation vectors and solving the resulting eigenvalue problem, the formulation extracted principal spatial deformation modes. The researchers retained the first four geometric modes, which conserved 98.5 percent of the total modal energy while halving the effective design space dimensionality from eight parameters to four latent modal coefficients. This linear reduction technique enabled the surrogate models to establish accurate input-output relationships across diverse blade profiles despite the complete absence of physical meaning in the projected modal coordinates.

The physical test case evaluated a low-Reynolds-number linear outlet guide vane cascade originally investigated experimentally in the transonic cascade wind tunnel of the German Aerospace Center in Cologne. The multi-objective optimization problem sought to simultaneously minimize the aerodynamic total pressure loss coefficient at the nominal design operating point and the average total pressure loss coefficient across two off-design incidence conditions representing near-stall and near-choking states, subject to manufacturing constraints on thickness, blade area, center of gravity, and edge radii. Both fidelity levels were solved using wolf, an unstructured vertex-centered Navier-Stokes solver. The low-fidelity model computed steady Reynolds-Averaged Navier-Stokes equations using the standard Spalart-Allmaras turbulence model on coarse meshes generated via gmsh containing sixteen thousand elements, completing in three hours. Conversely, the high-fidelity model utilized the transition-sensitive SA-BCM model coupled with feflo.a anisotropic metric-based mesh adaptation on Mach sensor fields, requiring forty-eight hours across ninety-six threads to resolve suction-side laminar separation bubbles.

Prior to deploying multi-fidelity surrogates, the authors conducted brute-force single-fidelity optimizations using fifty generations of NSGA-II, which revealed profound discrepancies between low- and high-fidelity Pareto fronts. The low-fidelity solver, failing to capture the laminar separation bubble, yielded tightly clustered candidates that exhibited an inverse correlation relative to high-fidelity objective spaces, confirming that uncorrected low-fidelity simulations would severely mislead optimization trajectories. Subsequently, six multi-fidelity configurations were tested across five independent runs to compare AR1 and neural network models with and without dimensionality reduction and infill regularization. Optimization performance was monitored after five and ten infill steps using Inverted Generational Distance, weakly Pareto-compliant Inverted Generational Distance Plus, and Hypervolume metrics calculated relative to the reference high-fidelity Pareto front, providing rigorous quantitative benchmarks of convergence speed and design space coverage under strict evaluation ceilings.

Quantitative evaluations demonstrated that after ten infill iterations, the Bayesian optimization strategy assisted by the AR1 co-kriging surrogate and Proper Orthogonal Decomposition dimensionality reduction achieved the best overall performance across all three multi-objective metrics. In the reduced four-dimensional latent space, Bayesian adaptive sampling rapidly explored the design boundaries, outperforming the neural network genetic algorithm framework. However, when optimization proceeded in the unreduced eight-dimensional parameter space, the performance advantage diminished, with both the AR1 and neural network strategies exhibiting comparable overall metrics. The regularized acquisition function yielded no measurable improvement in convergence speed after five infills and performed similarly to standard Bayesian sampling after ten steps in reduced dimensions, indicating that acquisition regularization depends heavily on problem dimensionality.

Final post-processing using NSGA-II on the converged surrogates generated dense predicted Pareto fronts, but the authors observed that predicted candidates situated far from high-fidelity training samples extended into physically unfeasible design spaces. Recomputing predicted optimal designs with the high-fidelity solver confirmed that surrogate predictions remained highly accurate near well-sampled regions but exhibited significant discrepancies in data-sparse zones, leading the authors to recommend mandatory high-fidelity recomputation for any candidate selected from surrogate-predicted fronts. Overall, the dimension-reduced Bayesian co-kriging framework successfully discovered realistic compressor blade geometries that improved nominal pressure loss by ten to seventeen percent and off-design losses by twenty-five to thirty percent, alongside balanced compromise profiles yielding simultaneous improvements of ten and twenty-two percent, establishing an efficient computational methodology for future large-eddy simulation workflows.

Subject of Research: Aerodynamic shape optimization of turbomachinery components using multi-fidelity surrogate models under extreme simulation cost imbalance

Article Title: Comparison of multi-fidelity surrogate models for multi-objective aerodynamic optimization in turbomachinery under extreme cost imbalance

Article References: Schouler, M., Belme, A., & Cinnella, P. (2025). Comparison of multi-fidelity surrogate models for multi-objective aerodynamic optimization in turbomachinery under extreme cost imbalance. Advanced Modeling and Simulation in Engineering Sciences, 12(1), Article 35. https://doi.org/10.1186/s40323-025-00316-3

Image Credits: AI Generated

DOI: 10.1186/s40323-025-00316-3

Keywords: aerodynamic shape optimization, turbomachinery, multi-fidelity surrogate modeling, Bayesian optimization, co-kriging, proper orthogonal decomposition, free-form deformation, neural networks, mesh adaptation, computational fluid dynamics, outlet guide vane, Spalart-Allmaras model, active learning, Pareto front, genetic algorithms, hypervolume